Page - 76 - in Document Image Processing

J. Imaging 2018,4, 37 ofoptimalalignments.Now, theDTWdistance isapproximatedas thesumof theEuclideandistances over theglobalprincipalalignments. FastApprxDTW(x1,x2)= ∑ π∈GX Euclidπ(x1,x2) (3) whereGX is the set of global principal alignments for the givendataX andEuclidπ(x1,x2) is the Euclideandistancebetweenx1 andx2 over thealignmentπ.Notice that theDTWdistancebetween twosamples is theEuclideandistance (grounddistance)over theoptimalalignment. Figure2.Thetopalignmentsbetweenfewsamples from2differentclasses.Here,X-axis is the length of thesamples fromclass1andY-axis is the lengthof thesamples fromclass2. ToshowtheperformanceofFastApprxDTW[20],wehavecomparedwithnaiveDTWdistanceand Euclideandistance forwordretrievalproblem.Here, thesedistancemeasuresareusedforcomparing wordimagerepresentations. Thedatasetcontains imagesfromthreedifferentwordclasses. Theresults aregiven inTable1.Nearestneighbor isusedforretrievingthesimilarsamples. Theperformance is measuredbymeanAveragePrecision (mAP).Fromtheresults,wecanobserve thatFastApprxDTWis comparable tonaiveDTWdistanceanditperformsbetter thanEuclideandistance. Table1.ThecomparisonoftheperformanceofDTWdistance,FastApprxDTWandEuclideandistance asasimilaritymeasure forawordretrievalproblem. DTWDistance FastApprxDTW Euclidean mAPscore 0.96 0.94 0.82 4.QuerySpecificFastDTWDistance InFastapproximateDTWdistance [20] (Section3), theglobalprincipalalignmentsarecomputed fromthegivendata.Here,noclass information isusedwhilecomputingthealignmentsandalso these alignmentsarequery independent, i.e., query information isnotusedwhile computing theglobal principalalignments. In thissection,weintroduceQueryspecificDTWdistance,which iscomputed usingqueryspecific (global)principal alignments. TheproposedQueryspecificDTWdistancehas beenfoundtogiveamuchbetterperformancewhenusedwith thedirectqueryclassifier. LetX bethegivendataandall thesamplesarescaledtoafixedsize. Let{C1,C2, . . . ,CN}bethe mostfrequentNclassesfromthedataandμ1, . . . ,μNbetheircorrespondingclassmeans. Thematching processusingthequeryspecificprincipalalignments isas follows: (i) Divide each sample from the frequent classes to a fixed number p of equal size portions. Let xi1, . . . ,xi|ci| be the samples (sequences) from the ith class ci, where |ci| is the number of samples in theclass ci. Thecutportions for theclassmeansμi aredenotedasμi1, . . . ,μ i p,where 76